Subgroups — #10621409

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	#66636	C₂	1	2	yes	maximal
3	C₄⋊Dic₂₁	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₂₁	C₂	1	2	yes	maximal, complemented
5	#6489	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	C₄	1	4	yes	complemented
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁	C₂²	1	4	yes	—
11	C₂²×C₄×C₇	D₃	1	1	yes	characteristic
12	C₂²⋊Dic₇	—	3	2	no	—
13	C₄⋊Dic₇	—	3	2	no	—
14	C₄×Dic₇	—	3	2	no	—
15	#926	—	7	1	no	maximal, Hall
16	C₂²×C₃×C₇	—	2	2	no	—
17	C₂²×C₃×C₇	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
21	Dic₂₁	—	2	4	no	—
22	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄×C₇	Dic₃	1	4	yes	complemented
24	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
26	C₂×Dic₇	—	3	4	no	—
27	C₂²×C₃×C₄	D₇	1	1	yes	characteristic
28	#217	—	7	2	no	—
29	C₄⋊Dic₃	—	7	2	no	—
30	C₄×Dic₃	—	7	2	no	—
31	C₂×C₃×C₇	—	2	2	no	—
32	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₃×C₇	#41	1	2	yes	—
34	#117	—	21	1	no	Sylow
35	C₂²×C₇	—	2	2	no	—
36	C₂²×C₇	C₂×Dic₃	1	2	yes	—
37	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
38	C₄×C₇	C₂×Dic₃	1	2	yes	—
39	C₄×C₇	C₂×Dic₃	1	2	yes	—
40	Dic₇	—	6	4	no	—
41	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
42	C₂×C₃×C₄	Dic₇	1	4	yes	complemented
43	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
45	C₂×Dic₃	—	7	4	no	—
46	C₃×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₄	D₂₁	1	1	yes	characteristic
48	C₄⋊C₄	—	21	2	no	—
49	C₂²⋊C₄	—	21	2	no	—
50	C₄²	—	21	2	no	—
51	C₂×C₇	—	2	2	no	—
52	C₂×C₇	#235	1	2	yes	—
53	C₂×C₇	C₂²×Dic₃	1	1	yes	characteristic
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₂×Dic₇	1	2	yes	—
56	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
57	C₃×C₄	C₂×Dic₇	1	2	yes	—
58	C₃×C₄	C₂×Dic₇	1	2	yes	—
59	Dic₃	—	14	4	no	—
60	C₂³	C₂×D₂₁	1	1	yes	characteristic
61	C₂×C₄	—	21	4	no	—
62	C₂×C₄	Dic₂₁	1	4	yes	complemented
63	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
64	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
65	C₇	#926	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	#1157	1	2	yes	—
68	C₂×C₃	C₂²×Dic₇	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	C₂×Dic₂₁	1	2	yes	—
71	C₂²	C₂²×D₂₁	1	1	yes	characteristic
72	C₄	—	42	4	no	—
73	C₄	C₂×Dic₂₁	1	2	yes	—
74	C₄	C₂×Dic₂₁	1	2	yes	—
75	C₃	#6489	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	2	no	—
77	C₂	#66728	1	2	yes	—
78	C₂	C₂²×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂²×C₄
Frattini	C₂²	C₂²×D₂₁
Fitting	C₂²×C₃×C₄×C₇	C₂
socle, cosocle	C₂²×C₃×C₇	C₂³
prime radical	C₂³×C₃×C₇	C₂²
hypercenter	C₂²×C₄	D₂₁
hypocenter	C₃×C₇	#117
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄⋊Dic₂₁	C₂	semidirect
4. C₄×Dic₂₁	C₂	semidirect
7. C₂×C₃×C₄×C₇	C₄	semidirect
23. C₂×C₄×C₇	Dic₃	semidirect
42. C₂×C₃×C₄	Dic₇	semidirect
46. C₃×C₇	#117	semidirect
62. C₂×C₄	Dic₂₁	semidirect
65. C₇	#926	semidirect
75. C₃	#6489	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₄×C₇	2
#66636	2
C₄⋊Dic₂₁	2
C₄×Dic₂₁	2
#6489	3
#926	7